Asymptotic Optimality of Certain Empirical Bayes Simultaneous Testing Procedures

摘要

This paper is concerned with the problem of simultaneous testing for n-component decisions. Under the specific statistical model, the n components share certain similarity. Thus, empirical Bayes approach is employed. We give a general formulation of this empirical Bayes decision problem with a specialization to the problem of selecting good Poisson populations. Three empirical Bayes methods are used to incorporate information from different sources for making a decision for each of the n components. They are: nonparametric empirical Bayes, parametric empirical Bayes and hierarchical empirical Bayes. For each of them, a corresponding empirical Bayes decision rule is proposed. The asymptotic optimality properties and the convergence rates of the three empirical Bayes rules are investigated. It is shown that each of the three empirical Bayes rules, the rate of convergence is at least of order 0(exp(-cn + ln n)) for some positive constant c, where the value of c varies depending on the empirical Bayes rule used. (KR)